solve-x-2-y-1-x-2-f-t-dx-x-3-1-x-1-y-f-x-

Question Number 126285 by bramlexs22 last updated on 19/Dec/20

\Rightarrow s o l v e x^{2} y = \int_{1}^{x^{2}} f (\sqrt{t}) d x + x^{3} + \frac{1}{x} + 1

y = f (x) = ?

Answered by liberty last updated on 19/Dec/20

\Rightarrow \frac{d}{d x} [x^{2} y] = \frac{d}{d x} [\int_{1}^{x^{2}} f (\sqrt{t}) d t + x^{3} + x^{- 1} + 1]

\Rightarrow 2 x y + x^{2} \frac{d y}{d x} = 2 x f (x) + 3 x^{2} - x^{- 2}

\Rightarrow 2 x y + x^{2} \frac{d y}{d x} = 2 x y + 3 x^{2} - x^{- 2}

\Rightarrow x^{2} \frac{d y}{d x} = 3 x^{2} - x^{- 2}

\Rightarrow \frac{d y}{d x} = 3 - x^{- 4}

\Rightarrow y = \int (3 - x^{- 4}) d x

\Rightarrow y = 3 x + \frac{1}{3 x^{3}} + C

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